In Haskell, numeric types are defined using a combination of type classes and data types. The standard numeric types in Haskell include integers, floating-point numbers, and rational numbers. Here is an overview of how these numeric types are defined:

Integers: The integer type in Haskell is called `Int`

. It represents whole numbers with no fractional part. The exact range of `Int`

depends on the underlying platform, but it is typically a fixed-size signed integer type.

Floating-point numbers: Haskell provides the `Float`

and `Double`

types for representing floating-point numbers. `Float`

is a 32-bit floating-point type, while `Double`

is a 64-bit floating-point type. Floating-point numbers can represent both integer and fractional values with limited precision.

Rational numbers: Haskell offers the `Rational`

type for representing exact fractions. A `Rational`

value is a ratio of two `Integer`

numbers, representing the numerator and denominator of the fraction. This type allows for arbitrary levels of precision, but may carry a performance cost compared to floating-point numbers.

Type classes: Haskell uses type classes to define the behavior and operations that are applicable to numeric types. The key numeric type classes in Haskell include `Num`

, `Integral`

, `Fractional`

, and `RealFrac`

, among others. These type classes provide a set of common operations such as addition, subtraction, multiplication, and conversion between different numeric types.

Overloading: Haskell uses type inference and overloading to determine the appropriate implementation of numeric operations based on the context. This means that numeric operations work on different numeric types as long as they satisfy the required type class constraints. For example, the `+`

operator can be used to add two `Int`

values or two `Double`

values seamlessly.

Custom types: In addition to the standard numeric types, Haskell allows users to define their own numeric types and make them instances of the relevant type classes. This is particularly useful when working with domain-specific numeric concepts or specialized numeric representations.

Overall, Haskell provides a flexible and powerful system for defining and working with numeric types, allowing for both standard and customized numeric representations with shared operations and behaviors based on type classes.

## What is the default implementation for numeric types in Haskell?

The default implementation for numeric types in Haskell is specific to each type. Here are some examples:

- For integral types (e.g., Int and Integer), the default implementation includes common operations such as addition (+), subtraction (-), multiplication (*), division (div), and modulus (mod).
- For fractional types (e.g., Float and Double), the default implementation includes common operations such as addition (+), subtraction (-), multiplication (*), division (/), and square root (sqrt).
- For other numeric types (e.g., Word, Int8, Word16, etc.), the default implementation provides similar basic arithmetic operations that are appropriate for the specific type.

In general, the default implementation follows the expected behavior of the respective numeric operations based on mathematical rules and conventions.

## How are complex types defined in Haskell?

Complex types in Haskell can be defined using data type declarations. A complex type is typically composed of constructor functions and associated type signatures.

Here is an example of how complex types can be defined in Haskell:

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data Person = Person String Int name :: Person -> String name (Person n _) = n age :: Person -> Int age (Person _ a) = a |

In this example, `Person`

is a complex type that consists of a constructor function with two fields: a `String`

representing the person's name and an `Int`

representing their age. The `name`

and `age`

functions are defined to extract specific fields from a `Person`

.

Complex types can also have multiple constructors, allowing for more specialized instances. Here is an example with multiple constructors:

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data Shape = Circle Float | Rectangle Float Float area :: Shape -> Float area (Circle r) = pi * r * r area (Rectangle w h) = w * h |

In this case, `Shape`

can either be a `Circle`

with a single `Float`

parameter representing its radius or a `Rectangle`

with two `Float`

parameters representing its width and height. The `area`

function calculates the area based on the shape type.

This is a simplified introduction to defining complex types in Haskell, but it demonstrates the basic concepts involved.

## How are type constraints applied to numeric types in Haskell?

In Haskell, type constraints are applied to numeric types using the `Num`

type class. The `Num`

class provides a common interface for numeric operations such as addition, subtraction, multiplication, and others.

Numeric types that are instances of the `Num`

class satisfy certain constraints defined by the class. These constraints include the ability to perform arithmetic operations and to convert between different numeric types. For example, the following code demonstrates how the `Num`

class is used to apply type constraints to numeric types:

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add :: Num a => a -> a -> a add x y = x + y |

In this example, the type constraint `Num a =>`

specifies that the function `add`

can only be used with numeric types that are instances of the `Num`

class. The `+`

operator is also defined in the `Num`

class, so it can be used to perform addition on the values of type `a`

.

By using type constraints, Haskell allows developers to write more polymorphic code that can operate on a wider range of numeric types, as long as they satisfy the constraints defined by the `Num`

class. This provides a level of flexibility and abstraction in working with different numeric types in Haskell.

## How are numeric literals represented in Haskell?

In Haskell, numeric literals can be represented in several ways depending on the type of number:

**Integer literals**: Integer literals can be directly written as a sequence of digits without any decimal point or exponent. For example, 42, -123, or 1000000 are all valid integer literals.**Floating-point literals**: Floating-point literals can be written in two forms: decimal or scientific notation. Decimal notation is written as a sequence of digits, optionally including a decimal point and/or exponent. For example, 3.14, -0.25, or 100.0 are all valid floating-point literals. Scientific notation is written as a decimal literal, followed by the letter 'e' or 'E', and then an exponent. For example, 1.23e4 represents 1.23 multiplied by 10^4.**Rational literals**: Rational literals are written as a fraction using the form a % b, where a and b are integer literals. For example, 1 % 2 represents the fraction 1/2.**Complex literals**: Complex literals are written using the :+ operator and two floating-point literals. The first number represents the real part, and the second number represents the imaginary part. For example, (3.0 :+ 4.0) represents the complex number 3 + 4i.

These are the common ways to represent numeric literals in Haskell, but there are also additional types for representing numbers such as binary literals or characters for representing octal and hexadecimal literals.

## What are the different boolean types available in Haskell?

In Haskell, the `Bool`

type represents Boolean values and has two possible values:

**True**: Represents truth.**False**: Represents falsehood.

These values are used in logical operations and conditions within Haskell programs.